Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 67
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.72e10·2-s − 1.09e16·3-s + 1.48e20·4-s + 4.93e23·5-s + 1.89e26·6-s + 1.79e28·7-s − 1.51e28·8-s + 2.81e31·9-s − 8.48e33·10-s + 5.33e34·11-s − 1.63e36·12-s + 2.30e37·13-s − 3.08e38·14-s − 5.42e39·15-s − 2.16e40·16-s − 1.47e40·17-s − 4.85e41·18-s + 5.15e42·19-s + 7.32e43·20-s − 1.97e44·21-s − 9.17e44·22-s − 8.82e44·23-s + 1.66e44·24-s + 1.75e47·25-s − 3.96e47·26-s + 7.09e47·27-s + 2.66e48·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.14·3-s + 1.00·4-s + 1.89·5-s + 1.61·6-s + 0.876·7-s − 0.00846·8-s + 0.304·9-s − 2.68·10-s + 0.691·11-s − 1.14·12-s + 1.10·13-s − 1.24·14-s − 2.16·15-s − 0.993·16-s − 0.0886·17-s − 0.430·18-s + 0.747·19-s + 1.90·20-s − 1.00·21-s − 0.980·22-s − 0.212·23-s + 0.00967·24-s + 2.59·25-s − 1.57·26-s + 0.794·27-s + 0.882·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(67\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1,\ (\ :67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(1.108872104\)
\(L(\frac12)\)  \(\approx\)  \(1.108872104\)
\(L(\frac{69}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 1.72e10T + 1.47e20T^{2} \)
3 \( 1 + 1.09e16T + 9.27e31T^{2} \)
5 \( 1 - 4.93e23T + 6.77e46T^{2} \)
7 \( 1 - 1.79e28T + 4.18e56T^{2} \)
11 \( 1 - 5.33e34T + 5.93e69T^{2} \)
13 \( 1 - 2.30e37T + 4.30e74T^{2} \)
17 \( 1 + 1.47e40T + 2.75e82T^{2} \)
19 \( 1 - 5.15e42T + 4.74e85T^{2} \)
23 \( 1 + 8.82e44T + 1.72e91T^{2} \)
29 \( 1 + 1.08e49T + 9.56e97T^{2} \)
31 \( 1 - 8.83e49T + 8.34e99T^{2} \)
37 \( 1 + 4.39e52T + 1.17e105T^{2} \)
41 \( 1 + 2.89e53T + 1.13e108T^{2} \)
43 \( 1 + 3.62e54T + 2.76e109T^{2} \)
47 \( 1 + 8.12e54T + 1.07e112T^{2} \)
53 \( 1 - 7.18e57T + 3.36e115T^{2} \)
59 \( 1 - 5.28e58T + 4.43e118T^{2} \)
61 \( 1 - 5.02e59T + 4.14e119T^{2} \)
67 \( 1 + 3.52e60T + 2.22e122T^{2} \)
71 \( 1 - 1.01e62T + 1.08e124T^{2} \)
73 \( 1 + 1.36e62T + 6.96e124T^{2} \)
79 \( 1 - 4.98e63T + 1.38e127T^{2} \)
83 \( 1 + 4.78e63T + 3.78e128T^{2} \)
89 \( 1 + 1.28e64T + 4.06e130T^{2} \)
97 \( 1 + 4.06e66T + 1.29e133T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.57979973726182226728079958653, −16.68957707003960585558422663426, −13.79891076788364301035235451061, −11.31944854053093170157121852812, −10.13960623277719405287494624857, −8.778273531595656465865546320133, −6.54311877586695736437341601593, −5.29599700427440966724136933913, −1.78866306257034809070773597280, −0.971299662601215759494365511330, 0.971299662601215759494365511330, 1.78866306257034809070773597280, 5.29599700427440966724136933913, 6.54311877586695736437341601593, 8.778273531595656465865546320133, 10.13960623277719405287494624857, 11.31944854053093170157121852812, 13.79891076788364301035235451061, 16.68957707003960585558422663426, 17.57979973726182226728079958653

Graph of the $Z$-function along the critical line